Some Asymptotic Results in a Model of Population Growth II. Positive Recurrent Chains

We treat a model describing the continued formation and growth of mutant biological populations. At each transition time of a Poisson process a new mutant population begins its evolution with a fixed number of elements and evolves according to the laws of a continuous time positive recurrent Markov Chain Y (t) with stationary transition probabilities P₈₊ (t), i, k = 0, 1, 2, , t 0. Our principal concern is the asymptotic behavior of moments and of the distribution function of the functional S (t) = \number of different sizes of mutant populations at time t\. When the recurrence time distribution to any state of the Markov Chain Y (t) has a finite second moment, the moments of S (t) and limit behavior of its distribution function are controlled by the stationary measure associated with Y (t). When the second moment of the recurrence time distribution is infinite, then a local limit theorem and speed of convergence estimate for P₈₊ (t) with k = k (t), t are required to establish asymptotic formulas for moments of S (t).